Let R be a ring with identity and M be a right unitary R-module. In this paper we
introduce the notion of strongly coretractable modules. Some basic properties of this
class of modules are investigated and some relationships between these modules and
other related concepts are introduced. 

Let R be associative ring with identity and M is a non- zero unitary left module over R. M is called M- hollow if every maximal submodule of M is small submodule of M. In this paper we study the properties of this kind of modules.

    In this paper, we formulate and study a new property, namely indeterminacy (neutrosophic) of the hollow module. We mean indeterminacy hollow module is neutrosophic hollow module B (shortly Ne(B)) such that it is not possible to specify the conditions for satisfying it.  Some concepts have been studied and introduced, for instance, the indeterminacy local module, indeterminacy divisible module, indeterminacy indecomposable module and indeterminacy hollow-lifting module. Also, we investigate that if Ne(B) is an indeterminacy divisible module with no indeterminacy zero divisors, then any indeterminacy submodule Ne(K) of Ne(B) is an indeterminacy hollow module. Further, we study the relationship between the indeterminacy of hollow an

       A submodule N of a module M  is said to be s-essential if it has nonzero intersection with any nonzero small submodule in M. In this article, we introduce and study a class of modules in which all its nonzero endomorphisms have non-s-essential kernels, named, strongly -nonsigular. We investigate some properties of strongly -nonsigular modules. Direct summand, direct sums and some connections of such modules are discussed.        

  Let R be a commutative ring with unity. In this paper we introduce and study the concept of strongly essentially quasi-Dedekind module as a generalization of essentially quasiDedekind module. A unitary R-module M is called a strongly essentially quasi-Dedekind module if ( , ) 0 Hom M N M for all semiessential submodules N of M. Where a submodule N  of  an R-module  M  is called semiessential if , 0  pN for all nonzero prime submodules  P of  M .
 

Let be an associative ring with identity and let be a unitary left -module. Let  be a non-zero submodule of .We say that  is a semi- - hollow module if for every submodule  of  such that  is a semi- - small submodule ( ). In addition, we say that  is a semi- - lifting module if for every submodule  of , there exists a direct summand  of  and  such that  

The main purpose of this work was to develop the properties of these classes of module.

A submodule Ϝ of an R-module Ε is called small in Ε if whenever  , for some submodule W of Ε , implies  . In this paper , we introduce the notion of Ζ-small submodule , where a proper submodule Ϝ of an R-module Ε is said to be Ζ-small in Ε if  , such that  , then  , where  is the second singular submodule of Ε . We give some properties of Ζ-small submodules . Moreover , by using this concept , we generalize the notions of hollow modules , supplement submodules, and supplemented modules into Ζ-hollow modules, Ζ-supplement submodules, and Ζ-supplemented modules. We study these concepts and provide some of their relations .

         The main goal of this paper is to give  a new generalizations for two important classes in the category of modules, namely the class of small submodules and the class of hollow modules. They are purely small submodules and purely hollow modules respectively. Various properties of these classes of modules are investigated. The relationship between purely small submodules and P-small submodules which is introduced by Hadi and Ibrahim, is studied. Moreover, another characterization of  purely hollow modules is considered.

     In this paper we introduce the notions of t-stable extending and strongly t-stable extending modules. We investigate properties and characterizations of each of these concepts. It is shown that a direct sum of t-stable extending modules is t-stable extending while with certain conditions a direct sum of strongly t-stable extending is strongly t-stable extending. Also, it is proved that under certain condition, a stable submodule of t-stable extending (strongly t-stable extending) inherits the property.

In this paper, we introduce and study the concepts of hollow – J–lifting modules and FI – hollow – J–lifting modules as a proper generalization of both hollow–lifting and J–lifting modules . We call an R–module M as hollow – J – lifting if for every submodule N of M with is hollow, there exists a submodule K of M such that M = K Ḱ and K N in M . Several characterizations and properties of hollow –J–lifting modules are obtained . Modules related to hollow – J–lifting modules are  given .